Showing papers in "Transactions of the American Mathematical Society in 1989"

Multiresolution approximations and wavelet orthonormal bases of L^2(R)

Abstract: A multiresolution approximation is a sequence of embedded vector spaces   V j  jmember Z for approximating L 2 (R) functions. We study the properties of a multiresolution approximation and prove that it is characterized by a 2π periodic function which is further described. From any multiresolution approximation, we can derive a function ψ(x) called a wavelet such that   √  2 j ψ(2 j x −k)   (k ,j)member Z 2 is an orthonormal basis of L 2 (R). This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space H s .

Differential-difference operators associated to reflection groups

Abstract: There is a theory of spherical harmonics for measures invariant under a finite reflection group. The measures are products of powers of linear functions, whose zero-sets are the mirrors of the reflections in the group, times the rotation-invariant measure on the unit sphere in Rn . A commutative set of differential-difference operators, each homogeneous of degree -1, is the analogue of the set of first-order partial derivatives in the ordinary theory of spherical harmonics. In the case of R2 and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions. The analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions. This paper is part of an ongoing program to establish a workable theory for one particular class. The underlying structure is based on finite Coxeter groups: these are finite groups acting on Euclidean space, generated by reflections in the zero sets of a collection of linear functions (the "roots"); the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere. In addition, the weight function is required to be invariant under the action of the group. The resulting theory has strong similarities to the theory of spherical harmonics; this was established in previous papers of the author [3, 4, 5]. Most notably, a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second-order differential-difference operator. Ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions; the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials. In this paper we construct a commutative set of first-order differentialdifference operators associated to the second-order operator previously mentioned. Received by the editors June 1, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 33A45, 33A65, 20H15; Secondary 20C30, 42C10, 51F15.

Remarks on classical invariant theory

Abstract: A uniform formulation, applying to all classical groups simultaneously, of the First Fundamental Theory of Classical Invariant Theory is given in terms of the Weyl algebra. The formulation also allows skew-symmetric as well as symmetric variables. Examples illustrate the scope of this formulation.

Braids, link polynomials and a new algebra

Abstract: A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on 2 parameters. The decomposition of the corresponding algebras into irreducible components is given and it is shown how they are related to Jones' algebras and to Brauer's centralizer algebras. In [J,3] Vaughan Jones announced the discovery of a new polynomial invariant of knots and links, which bore many similarities to the classical Alexander polynomial, but was seen to detect properties of a link which could not be detected by the Alexander invariants. The discovery was a real surprise, one of those exciting moments in mathematics when two seemingly unrelated disciplines turn out to have deep interconnections. The discovery came about in the following way. Jones' earlier contributions in the area of Operator Algebras had produced, in [J,1], a family of algebras An(t), t E C, indexed by the natural numbers n = 1 , 2, 3, . .. , and equipped with a trace function T: An(t) -C. His algebra An(t) was a quotient of the well-known Hecke algebra of the symmetric group, which we denote by 2 (1, m) to delineate our particular 2-parameter version of it. Jones had discovered, in [J,2], that there were representations of Artin's braid group Bn in the algebra An(t), in fact there were maps Bn 0 Atn(m A(t) from Bn into the multiplicative group of An (t) which factored through Links enter the picture via braids. Each oriented link L in oriented S3 can be represented by a (nonunique) element fl in some braid group Bn. There is an equivalence relation on B = H0= Bn known as Markov equivalence, which determines a 1-1 correspondence between equivalence classes [fl] E B and isotopy types of the associated oriented links Lfl. Jones' discovery was that with a small renormalization his trace function on A o(t) = Hln=l An(t) could be made into a function which lifted to an invariant on Markov classes Received by the editors December 9, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25; Secondary 20F29, 20C07. The work of the first author was supported in part by NSF grant #DMS-8503758. The work of the second author was supported in part by NSF grant #DMS-8510816. ? 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page

Isometric dilations for infinite sequences of noncommuting operators

Abstract: This paper develops a dilation theory for {T,}n=l an infinite sequence of noncommuting operators on a Hilbert space, when the matrix [T1, T2, ... ] is a contraction. A Wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for { Tn }I are obtained. Some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given. This results are used to extend the Sz.-Nagy-Foia$ lifting theorem to this noncommutative setting. This paper is a continuation of [5] and develops a dilation theory for an infinite sequence { Tn }n= of noncommuting operators on a Hilbert space ' when En=1 Tn~Tn? < I, (I, is the identity on S). Many of the results and techniques in dilation theory for one operator [8] and also for two operators [3, 4] are extended to this setting. First we extend Wold decomposition [8, 4] to 'the case of an infinite sequence { Vn }%I of isometries with orthogonal final spaces. In ?2 we obtain a minimal isometric dilation for {Tn}% 1 by extending the Schaffer construction in [6, 4]. Using these results we give some theorems on the geometric structure of the space of the minimal isometric dilation. Finally, we give some sufficient conditions on a sequence { T7 }n to be simultaneously quasi-similar to a sequence {Rn n=I of isometries on a Hilbert space X% with En= RnRn n IX . In ?3 we use the above-mentioned theorems to obtain the Sz.-Nagy-Foia5 lifting theorem [7, 8, 1, 4] in our setting. In a subsequent paper we will use the results of this paper for studying the "characteristic function" associated to a sequence { Tn }lo 1 with EZ= 1 T T* < I . Throughout this paper A stands for the set { 1 , 2, ... , k} (k E N) or the set N={1,2,...}. Received by the editors September 1, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 47A20; Secondary 47A45.

Variational problems on contact Riemannian manifolds

Abstract: We define the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable CR manifolds. Then the torsion and the generalized Tanaka-Webster scalar curvature are defined properly. Furthermore, we define gauge transformations of contact Riemannian structure, and obtain an invariant under such transformations. Concerning the integral related to the invariant, we define a functional and study its first and second variational formulas. As an example, we study this functional on the unit sphere as a standard contact manifold.

Invariants of graphs in three-space

Abstract: Par association d'une collection de nœuds et d'aretes a un graphe dans un espace tridimensionnel, on obtient des invariants calculables du type de plongement du graphe. On considere deux types d'isotopie

Bifurcation of critical periods for plane vector fields

Abstract: On etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l'origine pour toutes les valeurs d'un parametre λ∈R N . On definit la fonction periode: (ξ,λ)→P(ξ,λ). On determine les points critiques de cette fonction pres du centre avec λ comme parametre de bifurcation

The nonlinear geometry of linear programming. I. Affine and projective scaling trajectories

Abstract: This series of papers studies a geometric structure underlying Karmarkar's projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure studied is the set of trajectories obtained by integrating this vector field, which we call Ptrajectories. We also study a related vector field, the affine scaling vector field, and its associated trajectories, called A-trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. Affine and projective scaling vector fields are each defined for linear programs of a special form, called strict standard form and canonical form, respectively. This paper derives basic properties of P-trajectories and A-trajectones. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for P-trajectories and A-trajectories. It shows that projective transformations map P-trajectories into P-trajectories. It presents Karmarkar's interpretation of A-trajectories as steepest descent paths of the objective function (c, x) with respect to the Riemannian geometry ds2 Z?= dx, dx, /x2 restricted to the relative interior of the polytope of feasible solutions. P-trajectories of a canonical form linear program are radial projections of A-trajectories of an associated standard form linear program. As a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program: This algorithm is essentially Karmarkar's algorithm. These trajectories are studied in subsequent papers by two nonlinear changes of variables called Legendre transform coordinates and projective Legendre transform coordinates, respectively. It will be shown that P-trajectories have an algebraic and a geometric interpretation. They are algebraic curves, and they are geodesics (actually distinguished chords) of a geometry isometric to a Hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions. The A-trajectories of strict standard form linear programs have similar interpretations: They are algebraic curves, and are geodesics of a geometry isometric to Euclidean geometry. Received by the editors July 28, 1986 and, in revised form, September 28, 1987 and March 21, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 90C05; Secondary 52A40, 34A34. Research of the first author partially supported by ONR contract N00014-87-K0214. (D 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page

On the Cauchy problem and initial traces for a degenerate parabolic equation

Abstract: The authors study the Cauchy problem for the degenerate parabolic equation ut = div(|Du| p−2 Du)(p<2), and find sufficient conditions on the initial trace u0 (and in particular on its behaviour as |x|→∞) for existence of a solution in some strip RN × (0,T). Using a Harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions. Uniqueness of solutions is shown if u0 belongs to L1loc(RN), but is left open in the case that u0 is merely a locally bounded measure. The results are closely related to papers by Aronson-Caffarelli, Benilan-Crandall-Pierre, and Dahlberg-Kenig about the porous medium equation ut = Δum. The proofs are different and allow one to generalize some of the above results to equations with variable coefficients.

On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains

Abstract: In this paper we study the first initial-boundary value problem for ut = Au + uP in conical domains D = (0, oo) x Q c RN where Q c SN-I is an open connected manifold with boundary. We obtain some extensions of some old results of Fujita, who considered the case D = RN . Let A = -ywhere y_ is the negative root of y(y + N 2) = wI and where wI is the smallest Dirichlet eigenvalue of the Laplace-Beltrami operator on Q. We prove: If 1 3, arbitrary otherwise) there are singular stationary solutions u,. If u(x, 0) 1 + 2/N . We obtain some related results for ut = Au + IxI uP in the cone.

The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories

Abstract: Karmarkar's projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem. This paper studies the set of trajectories obtained by integrating this vector field, called P-trajectories, as well as a related set of trajectories, called A-trajectories. The A-trajectories arise from another linear programming algorithm, the affine scaling algorithm. The affine and projective scaling vector fields are each defined for linear programs of a special form, called standard form and canonical form, respectively. These trajectories are studied using a nonlinear change of variables called Legendre transform coordinates, which is a projection of the gradient of a logarithmic barrier function. The Legendre transform coordinate mapping is given by rational functions, and its inverse mapping is algebraic. It depends only on the constraints of the linear program, and is a one-to-one mapping for canonical form linear programs. When the polytope of feasible solutions is bounded, there is a unique point mapping to zero, called the center. The A-trajectories of standard form linear programs are linearized by the Legendre transform coordinate mapping. When the polytope of feasible solutions is bounded, they are the complete set of geodesics of a Riemannian geometry isometric to Euclidean geometry. Each A-trajectory is part of a real algebraic curve. Each P-trajectory for a canonical form linear program lies in a plane in Legendre transform coordinates. The P-trajectory through 0 in Legendre transform coordinates, called the central P-trajectory, is part of a straight line, and is contained in the A-trajectory through 0, called the central A-trajectory. Each P-trajectory is part of a real algebraic curve. The central A-trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form (c, x) = ,u . It is also the set of minima of a parametrized family of logarithmic barrier functions. Power-series expansions are derived for the central A-trajectory, which is also the central P-trajectory. These power-series have a simple recursive form and are useful in developing "higher-order" analogues of Karmarkar's algorithm. A-trajectories are defined for a general linear program. Using this definition, it is shown that the limit point x,0 of a central A-trajectory on the boundary of the feasible solution polytope P is the center of the unique face of P containing x,0 in its relative interior. Received by the editors October 8, 1986 and, in revised form, June 9, 1987 and March 25, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 90C05; Secondary 52A40, 34A34. The first author was partially supported by ONR contract N00014-87-K0214. ( 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page

Realizing rotation vectors for torus homeomorphisms

Abstract: We consider the rotation set p(F) for a lift F of a homeomorphism f: T2 -_ T2, which is homotopic to the identity. Our main result is that if a vector v lies in the interior of p(F) and has both coordinates rational, then there is a periodic point x E T2 with the property that

Stability in the isoperimetric problem for convex or nearly spherical domains in ⁿ

Abstract: For convex bodies D in Rn the deviation d from spherical shape is estimated from above in terms of the (dimensionless) isoperimetric deficiency A of D as follows: d < f(A) (for A sufficiently small). Here f is an explicit elementary function vanishing continuously at 0. The estimate is sharp as regards the order of magnitude of f . The dimensions n = 2 and 3 present anomalies as to the form of f . In the planar case n = 2 the result is contained in an inequality due to T. Bonnesen. A qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies D in Rn in the sense that, as D varies, d -p 0 for A -A 0. The proof of the estimate d < f(A) is based on a related estimate in the case of domains (not necessarily convex) that are supposed a priori to be nearly spherical in a certain sense.

The connection matrix theory for Morse decompositions

Abstract: The connection matrix theory for Morse decompositions is introduced. The connection matrices are matrices of maps between the homology indices of the sets in the Morse decomposition. The connection matrices cover, in a natural way, the homology index braid of the Morse decomposition and provide information about the structure of the Morse decomposition. The existence of connection matrices of Morse decompositions is established, and examples illustrating applications of the connection matrix are provided.

Weighted norm inequalities for the continuous square function

Abstract: We prove new weighted norm inequalities for real-variable analogues of the Lusin area function. We apply our results to obtain new: (i) weighted norm inequalities for singular integral operators; (ii) weighted Sobolev inequalities; (iii) eigenvalue estimates for degenerate Schrodinger operators.

On infinite root systems

Abstract: We define in an axiomatic fashion the concept of a set of root data that generalizes the usual concept of root system of a Kac-Moody Lie algebra. We study these objects from a purely formal and geometrical point of view as well as in relation to their associated Lie algebras. This leads to a coherent theory of root systems, bases, subroot systems, Lie algebras defined by root data, and subalgebras.

Melnikov transforms, bernoulli bundles, and almost periodic perturbations

Abstract: In this paper we study nonlinear time-varying perturbations of an autonomous vector field in the plane R2 . We assume that the unperturbed equation, i.e. the given vector field has a homoclinic orbit and we present a gen- eralization of the Melnikov method which allows us to show that the perturbed equation has a transversal homoclinic trajectory. The key to our generalization is the concept of the Melnikov transform, which is a linear transformation on the space of perturbation functions. The appropriate dynamical setting for studying these perturbation is the concept of a skew product flow. The concept of transversality we require is best understood in this context. Under conditions whereby the perturbed equation admits a transversal homoclinic trajectory, we also study the dynamics of the perturbed vector field in the vicinity of this trajectory in the skew product flow. We show the dynamics near this trajectory can have the exotic behavior of the Bernoulli shift. The exact description of this dynamical phenomenon is in terms of a flow on a fiber bundle, which we call, the Bernoulli bundle. We allow all perturbations which are bounded and uniformly continuous in time. Thus our theory includes the classical periodic perturbations studied by Melnikov, quasi periodic and almost periodic perturbations, as well as toroidal perturbations which are close to quasi periodic perturbations.

On the reconstruction of topological spaces from their groups of homeomorphisms

Abstract: For various classes K of topological spaces we prove that if X\\ , X2 6 K and X\\ , X2 have isomorphic homeomorphism groups, then X\\ and X2 are homeomorphic. Let G denote a subgroup of the group of homeomorphisms H(X) of a topological space X. A class K of (X, G) 's is faithful if for every (X\\, G\\) , (X2, G2) € K , if (g) hgh~[ . Theorem 1: The following class is faithful: {{X, H(X)) | ( X is a locally finite-dimensional polyhedron in the metric or coherent topology or I is a Euclidean manifold with boundary) and for every x G X x is an accumulation point of (g(x)\\g 6 H(X)}} U {{X, G) \\ X is a differentiable or a PL-manifold and G contains the group of differentiable or piecewise linear homeomorphisms} U{(X, H(X)) | X is a manifold over a normed vector space over an ordered field}. This answers a question of Whittaker [W], who asked about the faithfulness of the class of Banach manifolds. Theorem 2: The following class is faithful: {(X, G) \\ X is a locally compact Hausdorff space and for every open T Ç X and x € T (g(x) | g 6 H(X) and g \\ (X T) = Id} is somewhere dense}. Note that this class includes Euclidean manifolds as well as products of compact connected Euclidean manifolds. Theorem 3: The following class is faithful: {(X, H(X)) | (1) A\" is a O-dimensional Hausdorff space; (2) for every x € X there is a regular open set whose boundary is {x} ; (3) for every x 6 X there are gi , g2 € G such that x ^ gi(x) ^ gi(x) ^ x , and (4) for every nonempty open V C X there is g e H(X) {Id} such that g Í (X V) = Id} . Note that (2) is satisfied by O-dimensional first countable spaces, by order topologies of linear orderings, and by normed vector spaces over fields different from R. Theorem 4: We prove (Theorem 2.23.1) that for an appropriate class KT of trees {(Aut(T), T;<, o,Op) | T € KT} is first-order interpretable in {Aut(7\") | T € KT} .

Equivariant Morse theory for starshaped Hamiltonian systems

Abstract: Soit ∑ une hypersurface en forme d'etoile dans R 2n ; le probleme de trouver des caracteristiques fermees de ∑ peut se reduire classiquement a un probleme variationnel. Ceci conduit a etudier une fonctionnelle S 1 -equivariante sur un espace de Hilbert

Summation, transformation, and expansion formulas for bibasic series

Abstract: Etude des extensions a deux bases de la formule de transformation d'Euler et de la formule de developpement de Fields-Wimp. Etablissement d'une formule de transformation a quatre bases independantes, d'une formule d'inversion de Lagrange et de quelques formules de sommation de degres 2, 3 et 4

Multiple solutions of perturbed superquadratic second order Hamiltonian systems

Abstract: : In this paper, the author proves the existence of infinitely many distinct T-periodic solutions of the perturbed second order Hamiltonian systems q + V'(q) = f(t) under the condition that V is C and superquadratic via minimax methods. We also obtain similar results for general nonautonomous second order Hamiltonian systems and perturbed Lagrangian systems. Keywords: invariance; Asymmetry; A priori growth estimates; multiple periodic solutions.

Injectivity of operator spaces

Abstract: We study the structure of injective operator spaces and the existence and uniqueness of the injective envelopes of operator spaces. We give an easy example of an injective operator space which is not completely isometric to any C*-algebra. This answers a question of Wittstock [23]. Furthermore, we show that an operator space E is injective if and only if there exists an injective C*-algebra A and two projections p and q in A such that E is completely isometric to pAq .

A criterion for the boundedness of singular integrals on hypersurfaces

Abstract: On donne des conditions geometriques sur une hypersurface de R n pour que certaines integrales singulieres sur cette hypersurface definissent des operateurs bornes sur L 2

Stability of viscous scalar shock fronts in several dimensions

Abstract: We prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions. The proof uses the "integrated equation" and an effective equation for the motion of the front itself. We derive energy estimates that balance terms from the integrated equation with terms from the front motion equation. In this paper we prove that viscous shock profiles for scalar conservation laws are stable in two or more space dimensions. These multidimensional stability questions are separate from their one dimensional analogues because of the possibility of transverse instabilities such as those that occur in combustion fronts (Lu) and in shock waves with phase changes. The proof here is a rigorous version of arguments that are used to derive effective equations (such as the Kuramoto-Sivashinsky equation) to describe the behavior of fronts. The one dimensional stability for scalar conservation laws was proven by II ' in and Oleinik (IO) using the "integrated equation" (see below) and a max- imum principle. Another proof, based on weighted norms and spectral theory for the linearized problem, was given by Sattinger (S). The multidimensional stability proof below has more in common with the stability proofs for systems of conservation laws in one space dimension begun by Kawashima and Mat- sumura (KM) and Goodman (Go) and completed by Liu (Li). These proofs use L2 energy estimates for the integrated equation. We consider equations of the form (!) ut + f(u)x + g(u)y=:uxx + uyy, where f(u) is a strictly convex function of u :

Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems

Abstract: Transfer operators and zeta functions of piecewise monotonie and of more general piecewise invertible dynamical systems are studied. To this end we construct Markov extensions of given systems, develop a kind of Fredholm theory for them, and carry the results back to the original systems. This yields e.g. bounds on the number of ergodic maximal measures or equilibrium states.

A new algebraic approach to microlocalization of filtered rings

Abstract: Using the construction of the Rees ring associated to a filtered ring we provide a description of the microlocalization of the filtered ring by using only purely algebraic techniques. The method yields an easy approach towards the study of exactness properties of the microlocalization functor. Every microlocalization at a regular multiplicative Ore set in the associated graded ring can be obtained as the completion of a localization at an Ore set of the filtered ring.

Periodic orbits of maps of Y

Abstract: We introduce some notions that are useful for studying the behavior of periodic orbits of maps of one-dimensional spaces. We use them to characterize the set of periods of periodic orbits for continuous maps of Y = (z e C: z3 e [0,1]} into itself having zero as a fixed point. We also obtain new proofs of some known results for maps of an interval into itself.

Global solvability of the derivative nonlinear Schrödinger equation

Abstract: The derivative nonlinear Schrodinger equation (DNLS) iqt = qxx ± (q*q2)x,

Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities

Abstract: We give conditions for the existence, uniqueness, and asymptotic stability of periodic solutions of a second-order differential equation with piecewise linear restoring and 27r-periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer. With suitable restrictions on the restoring and forcing in the undamped case, we give a necessary and sufficient condition.